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1 Observation of unconventional edge states in photonic graphene Yonatan Plotnik 1 *, Mikael C. Rechtsman 1 *, Daohong Song 2 *, Matthias Heinrich 3, Julia M. Zeuner 3, Stefan Nolte 3, Yaakov Lumer 1, Natalia Malkova 4, Jingjun Xu 2, Alexander Szameit 3, Zhigang Chen 2,4, Mordechai Segev 1 1 Technion - Israel Institute of Technology, Technion City 32000, Haifa, Israel 2 The Key Laboratory of Weak-Light Nonlinear Photonics, Ministry of Education and TEDA Applied Physical School, Nankai University, Tianjin , China 3 Institute of Applied Physics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, Jena, Germany 4 Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132, USA CONTENTS Experimental Setup... 2 Optically-induced photonic honeycomb lattices... 2 Photonic honeycomb lattices fabricated via femtosecond direct laser writing... 4 Extended tight binding derivation and the origin of the novel edge states... 5 Derivation of additional terms in coupled mode theory (tight-binding)... 5 Implementing the extended tight-binding model to understand the origin of the newly found edge state... 7 Relation to edge states in photonic crystals... 8 Bibliography... 9 NATURE MATERIALS 1

2 EXPERIMENTAL SETUP Optically-induced photonic honeycomb lattices Our first experimental setup (see Fig. S1) relies on the optical induction method 1,2 which is used to create a honeycomb waveguide array in a photorefractive strontium barium niobate (SBN) crystal. We examine the edge states by launching a tightly focused probe beam along the surface of the photonic lattice, and monitor its transverse intensity pattern exiting the lattice. To do that, we use a beam from an argon-ion laser operating at 488nm wavelength and split it into two beams, one for writing the waveguide array pattern into the crystal, and the other for probing the written structure. The writing beam passes through a rotating diffuser, turning into partially spatial incoherent, before it is sent through a specially designed amplitude mask 3. The mask generates three interfering beams that together generate a triangular lattice interference pattern. Such a pattern remains invariant during linear propagation through the crystal, except for slight deformation and diffraction at the edge of the lattice. To generate a honeycomb lattice with the desired edge structure, as illustrated in Fig. 2(a), we employ self-defocusing nonlinearity on the triangular intensity pattern 4 (Fig. 2(a), bright spots). Specifically, we apply a DC electric field parallel to the crystalline c-axis of the ferroelectric crystal in the direction perpendicular to the propagation axis, and set the polarity of the bias voltage such that it gives rise to a selfdefocusing nonlinearity 4. Under these conditions, the refractive index is lowered in regions where the intensity is high, and consequently the triangular interference pattern is transformed into its "negative": a honeycomb pattern of waveguides 4 (Fig. 2(a), blue spots), in the same structure as the insert in Fig. S1. In this way, we establish an array of waveguides residing in the gaps between regions of high intensity, a honeycomb lattice with sharp edges that remain invariant during propagation throughout the crystal. The probe beam is affected by the induced refractive index pattern (lattice) and propagates under the influence thereof. In the experiment, the probe beam is cylindrically focused into a narrow stripe beam along edge of the lattice, and is launched onto the edge waveguides. This method allows us to probe the armchair and the zigzag edges of the induced honeycomb lattice. 2 NATURE MATERIALS

3 SUPPLEMENTARY INFORMATION Figure S1: Experimental setup for generating the optically-induced honeycomb photonic lattice (top-right insert), and for probing the lattice edges with an appropriately oriented stripe beam. NATURE MATERIALS 3

4 Photonic honeycomb lattices fabricated via femtosecond direct laser writing Our second experimental setup (Fig. S2) relies on employing a honeycomb arrangement of waveguides written using the femtosecond direct laser writing technique in fused silica 5. This technique facilitates very sharp edges, which are crucial specifically for testing the dispersion properties of the edge states (Fig. 1(c) in the paper). To probe the edge states and their dependence on the launch angle (transverse momentum), we launch a beam on both zigzag and bearded edges, while controlling the incident angle of the probe beam launched onto the input facet of the array. For probing both the zigzag and the bearded edges, we use an array that has one of each kind of edge on opposite sides of the array. We shape the probe beam using an adjustable rectangular slit, and image its Fourier transform (with appropriate demagnification) on a rotatable mirror. Using a 4-f system (two identical lenses at a distance of 2f from one another), we image the beam from the face of the mirror onto the face of the fused silica sample. This allows us to control the input angle of the beam by rotating the mirror, while maintaining its shape. By controlling the input angle, we control the transverse wavevector, k x, of the input beam and are able to scan the through transverse wavevector in search of edge confinement due to the existence of edge states. Figure S2: Experimental setup for probing the dispersion properties of edge states in the honeycomb photonic lattice written in fused silica 4 NATURE MATERIALS

5 SUPPLEMENTARY INFORMATION EXTENDED TIGHT BINDING DERIVATION AND THE ORIGIN OF THE NOVEL EDGE STATES Derivation of additional terms in coupled mode theory (tight-binding) In this Appendix, we elaborate on the derivation of the additional terms in the tight-binding scheme and investigate their origin. The tight-binding scheme enables writing a Hamiltonian of a system in a basis composed of the bound states of individual potential wells, which in a photonic lattice correspond to optical waveguides. For the case described in our paper, the system is a laser beam propagating paraxially in a waveguide array, but our analysis also applies to electrons (where spin is neglected) in an array of potential wells (like the potentials generated by nuclei in a solid). We start by defining to be the potential well of site n, and as the ground state eigenmode of the Schrödinger equation within that potential, i.e. (S1) Where is the spatial Laplacian in two dimensions. The wavefunctions will be the basis functions for our Hamiltonian matrix, so let us define. By limiting ourselves to this basis, we are making the first approximation of the tight binding model. In most cases, this is an excellent approximation, as is the case considered here. Our goal is to find the matrix operator that determines the time evolution of the coefficients in the subspace spanned by this limited basis. The Schrödinger equation describing the whole lattice of potential wells is: (S2) Substituting the definition of into equation (S1), and using equation (S2), we find. (S3) We multiply (S3) from the left with and get (S4) NATURE MATERIALS 5

6 We can write this in matrix form: (S5) Where we used the definition. Finally, we multiply on the left by : (S6) Let us now interpret the different terms as labeled in (S4) and consider reasonable approximations that can be made. The matrix S is the overlap matrix between the different basis modes. The diagonal entries of this matrix are each unity, as the modes are normalized. In general, the off-diagonal terms are non-zero since the modes are not orthogonal, but they are significantly smaller than one, and thus can be neglected at first order and S becomes the identity matrix. The terms in C are the well-known coupling terms. The first approximation to these terms is made by taking into account only the on-site potential in calculating, and thus it becomes. A second approximation is to assume that is small for nonneighboring sites and thus can be taken to be zero for sites far enough away from each other. The last term to discuss is, which is the correction to the on-site energy due to the potential of the neighboring sites. The terms of this diagonal matrix are usually much smaller than the terms in the coupling matrix, and are thus usually neglected. Moreover, in a periodic or infinite lattice system, all the elements of are equal, and thus, if S is taken to be unity, can be taken to be zero (as it only shifts the energy of all the modes equally). In the cases where the system is a finite lattice, (namely, our system) the elements of representing the lattice edge sites may be significantly different than the lattice bulk sites, as they have a different number of neighbors. Specifically, the sites on the bearded edge of a honeycomb lattice have only 1 neighbor and, thus are significantly different than the bulk sites which have 3 neighbors, creating an effective defect on the lattice edge. 6 NATURE MATERIALS

7 SUPPLEMENTARY INFORMATION Implementing the extended tight-binding model to understand the origin of the newly found edge state Implementing this extended tight binding model in our honeycomb lattice, and going beyond the first order approximations, we find that while the overlap term S and non-nearest-neighbor couplings in C only distort the band structure but do not add new features to it, the diagonal term may add additional features. In infinite crystals or purely periodic structures, all the elements in are identical and can usually be ignored, but in a finite system, the elements of the edge sites are different than those of the bulk. This means that an effective-defect is formed at the array edge, and can cause a Tamm-like edge state to appear at k x =π/a. The issue of why the edge state appears at exactly k x =π/a merits further discussion. It has been shown rigorously 6 that very small edge perturbations (such as defects or dilations of the bond length) can cause edge states to emerge at points in the edge Brillouin zone with zero dispersion in the direction transverse to the edge (i.e., the values of k x for which ), where represents the bulk band structure. This is indeed the case at k x =π/a. It follows 6 that the total lack of dispersion perpendicular to the edge (due to this degeneracy) causes any edge perturbation to localize energy on the edge, rather than allowing it to disperse into the bulk. It can therefore be argued that the edge itself is acting as a sufficient perturbation to induce the edge state, but only near the van Hove singularity at k x =π/a, where the spatial dispersion perpendicular to the edge is extremely low. In other words, because the effective-defect is usually very small (compared to the coupling constants C), it significantly affects the band structure only at points where C does not, namely, at the van-hove singularities, where all modes are degenerate. At these points any slight edge defect affects the spectrum of the eigenmodes and gives rise to Tamm-like edge states. Numerical simulations of the extended tight binding model support this finding, because when a term is added to the tight binding Hamiltonian, the edge modes appear, and have the same features seen in the continuum simulations as shown in Fig. 4(i). NATURE MATERIALS 7

8 RELATION TO EDGE STATES IN PHOTONIC CRYSTALS Our experiments in 'photonic graphene' (honeycomb lattices of optical waveguides) call for an explanation on the relation to edge states in photonic crystals and photonic crystal slabs. Surface and edge physics in photonic systems have been studied previously in the context of photonic crystals 7 and lattices 8,9. Theoretical work has predicted edge states in photonic graphene 10,11,12 that terminate at Dirac cones, and experiments in the microwave regime have observed edge states in macroscopic graphene-like structures 13,14. However, those microwave experiments did not have momentum resolution, meaning the states could not be classified by their location in the Brillouin zone, hence many questions remained open and many assertions remained untested. In the current work, we address such questions, and most importantly we find a new type of edge state that has not been observed nor predicted as of yet. An additional point to be emphasized, as explained in the paper, is that the propagation of electromagnetic waves in paraxial photonic lattices is described by the paraxial wave equation, which is analogous to the Schrödinger equation. As such, there are significant differences between them and photonic crystals, which do not conform to the paraxial wave equation but to the full Maxwell equations. For example, in the context of waves in graphene-like structures, honeycomb photonic lattices display Dirac points just like carbon-based graphene does, with analogous features, whereas three-dimensional photonic crystals display Weyl points with very different properties 15. Likewise, the edge states in photonic crystals and photonic crystal slabs are different than those in 'photonic graphene' and of electronic carbon-based graphene, as they are not strictly confined to the lattice and radiate away since photonic crystal modes are never bound. To conclude, the effects measured in photonic crystal systems, while interesting, have little to do with the current work, and the differences between photonic crystals and waveguide arrays mean that physical effects of photonic crystal edge states cannot be carried over to electronic systems (e.g., graphene) because their behavior is governed by a different wave equation, while the results measured in a waveguide array system can be easily carried over to electronic systems as they are both governed by the Schrödinger equation. 8 NATURE MATERIALS

9 SUPPLEMENTARY INFORMATION BIBLIOGRAPHY 1. Efremidis, N. K., Sears, S., Christodoulides, D. N., Fleischer, J. W. & Segev, M. Discrete solitons in photorefractive optically induced photonic lattices. Phys. Rev. E 66, (2002). 2. Fleischer, J. W., Segev, M., Efremidis, N. K. & Christodoulides, D. N. Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature 422, (2003). 3. Martin, H., Eugenieva, E. D., Chen, Z. & Christodoulides, D. N. Discrete Solitons and Soliton-Induced Dislocations in Partially Coherent Photonic Lattices. Phys. Rev. Lett. 92, (2004). 4. Peleg, O. et al. Conical Diffraction and Gap Solitons in Honeycomb Photonic Lattices. Phys. Rev. Lett. 98, (2007). 5. Szameit, A. et al. Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica. Opt. Express 13, (2005). 6. Mattis, D. C. Electron states near surfaces of solids. Annals of Physics 113, (1978). 7. Joannopoulos, J. D., Johnson, S. G., Winn, J. N. & Meade, R. D. Photonic Crystals: Molding the Flow of Light (Second Edition). (Princeton University Press, 2011). 8. Makris, K. G., Suntsov, S., Christodoulides, D. N., Stegeman, G. I. & Hache, A. Discrete surface solitons. Opt. Lett. 30, (2005). 9. Malkova, N., Hromada, I., Wang, X., Bryant, G. & Chen, Z. Observation of optical Shockley-like surface states in photonic superlattices. Opt. Lett. 34, (2009). 10. Ochiai, T. & Onoda, M. Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states. Phys. Rev. B 80, (2009). 11. Ochiai, T. Topological properties of bulk and edge states in honeycomb lattice photonic crystals: the case of TE polarization. J. Phys.: Condens. Matter 22, (2010). 12. Jukić, D., Buljan, H., Lee, D.-H., Joannopoulos, J. D. & Soljačić, M. Flat photonic surface bands pinned between Dirac points. Opt. Lett. 37, (2012). 13. Bittner, S. et al. Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene. Phys. Rev. B 82, (2010). 14. Kuhl, U. et al. Dirac point and edge states in a microwave realization of tight-binding graphene-like structures. Phys. Rev. B 82, (2010). 15. Lu, L., Fu, L., Joannopoulos, J. D. & Soljačić, M. Weyl points and line nodes in gyroid photonic crystals. Nat Photon 7, (2013). NATURE MATERIALS 9